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High School Mathematics Geometry Vocabulary Word Wall Cards Table of Contents Reasoning, Lines, and Transformations Basics of Geometry 1 Basics of Geometry 2 Geometry Notation Logic Notation Set Notation Conditional Statement Converse Inverse Contrapositive Symbolic Representations Deductive Reasoning Inductive Reasoning Proof Properties of Congruence Law of Detachment Law of Syllogism Counterexample Perpendicular Lines Parallel Lines Skew Lines Transversal Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Consecutive Interior Angles Parallel Lines Midpoint Midpoint Formula Virginia Department of Education ©2013 Slope Formula Slope of Lines in Coordinate Plane Distance Formula Line Symmetry Point Symmetry Rotation (Origin) Reflection Translation Dilation Rotation (Point) Perpendicular Bisector Constructions: o A line segment congruent to a given line segment o Perpendicular bisector of a line segment o A perpendicular to a given line from a point not on the line o A perpendicular to a given line at a point on the line o A bisector of an angle o An angle congruent to a given angle o A line parallel to a given line through a point not on the given line o An equilateral triangle inscribed in a circle o A square inscribed in a circle o A regular hexagon inscribed in a circle o An inscribed circle of a triangle Geometry Vocabulary Cards Page 1 o A circumscribed circle of a triangle o A tangent line from a point outside a given circle to the circle Triangles Classifying Triangles by Sides Classifying Triangles by Angles Triangle Sum Theorem Exterior Angle Theorem Pythagorean Theorem Angle and Sides Relationships Triangle Inequality Theorem Congruent Triangles SSS Triangle Congruence Postulate SAS Triangle Congruence Postulate HL Right Triangle Congruence ASA Triangle Congruence Postulate AAS Triangle Congruence Theorem Similar Polygons Similar Triangles and Proportions AA Triangle Similarity Postulate SAS Triangle Similarity Theorem SSS Triangle Similarity Theorem Altitude of a Triangle Median of a Triangle Concurrency of Medians of a Triangle 30°-60°-90° Triangle Theorem 45°-45°-90° Triangle Theorem Geometric Mean Trigonometric Ratios Inverse Trigonometric Ratios Area of a Triangle Rectangle Rhombus Square Trapezoids Circle Circles Circle Equation Lines and Circles Secant Tangent Central Angle Measuring Arcs Arc Length Secants and Tangents Inscribed Angle Area of a Sector Inscribed Angle Theorem 1 Inscribed Angle Theorem 2 Inscribed Angle Theorem 3 Segments in a Circle Segments of Secants Theorem Segment of Secants and Tangents Theorem Three-Dimensional Figures Cone Cylinder Polyhedron Similar Solids Theorem Sphere Pyramid Polygons and Circles Polygon Exterior Angle Sum Theorem Polygon Interior Angle Sum Theorem Regular Polygon Properties of Parallelograms Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 2 Reasoning, Lines, and Transformations Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 3 Basics of Geometry Point – A point has no dimension. P It is a location on a plane. It is point P represented by a dot. Line – A line has one dimension. It is an infinite set of points represented by a line with two arrowheads that extends without end. m A B AB or BA or line m Plane – A plane has two dimensions extending without end. It is often represented by a parallelogram. plane ABC or plane N N A C B Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 4 Basics of Geometry Line segment – A line segment consists of two endpoints and all the points between them. A B AB or BA Ray – A ray has one endpoint and extends without end in one direction. C BC B Virginia Department of Education ©2013 Note: Name the endpoint first. BC and CB are different rays. Geometry Vocabulary Cards Page 5 Geometry Notation Symbols used to represent statements or operations in geometry. BC BC BC BC ∠ ABC m∠ ABC ∆ ABC || ⊥ ≅ ∼ Virginia Department of Education ©2013 segment BC ray BC line BC length of BC angle ABC measure of angle ABC triangle ABC is parallel to is perpendicular to is congruent to is similar to Geometry Vocabulary Cards Page 6 Logic Notation ⋁ ⋀ or and read “implies”, if… then… read “if and only if” read “if and only if” not ∴ therefore → ↔ iff ~ Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 7 Set Notation {} ∅ x| x: ⋃ ⋂ empty set, null set empty set, null set read “x such that” read “x such that” union, disjunction, or intersection, conjunction, and Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 8 Conditional Statement a logical argument consisting of a set of premises, hypothesis (p), and conclusion (q) hypothesis If an angle is a right angle, then its measure is 90°. conclusion Symbolically: if p, then q Virginia Department of Education ©2013 Geometry Vocabulary Cards p q Page 9 Converse formed by interchanging the hypothesis and conclusion of a conditional statement Conditional: If an angle is a right angle, then its measure is 90°. Converse: If an angle measures 90°, then the angle is a right angle. Symbolically: if q, then p Virginia Department of Education ©2013 Geometry Vocabulary Cards q p Page 10 Inverse formed by negating the hypothesis and conclusion of a conditional statement Conditional: If an angle is a right angle, then its measure is 90°. Inverse: If an angle is not a right angle, then its measure is not 90°. Symbolically: if ~p, then ~q Virginia Department of Education ©2013 Geometry Vocabulary Cards ~p~q Page 11 Contrapositive formed by interchanging and negating the hypothesis and conclusion of a conditional statement Conditional: If an angle is a right angle, then its measure is 90°. Converse: If an angle does not measure 90°, then the angle is not a right angle. Symbolically: if ~q, then ~p Virginia Department of Education ©2013 Geometry Vocabulary Cards ~q~p Page 12 Symbolic Representations Conditional if p, then q pq Converse if q, then p qp if not p, Inverse then not q if not q, Contrapositive then not p Virginia Department of Education ©2013 Geometry Vocabulary Cards ~p~q ~q~p Page 13 Deductive Reasoning method using logic to draw conclusions based upon definitions, postulates, and theorems Example: Prove (x ∙ y) ∙ z = (z ∙ y) ∙ x. Step 1: (x ∙ y) ∙ z = z ∙ (x ∙ y), using commutative property of multiplication. Step 2: = z ∙ (y ∙ x), using commutative property of multiplication. Step 3: = (z ∙ y) ∙ x, using associative property of multiplication. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 14 Inductive Reasoning method of drawing conclusions from a limited set of observations Example: Given a pattern, determine the rule for the pattern. Determine the next number in this sequence 1, 1, 2, 3, 5, 8, 13... Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 15 Proof a justification logically valid and based on initial assumptions, definitions, postulates, and theorems Example: Given: ∠1 ≅ ∠2 Prove: ∠2 ≅ ∠1 Statements ∠1 ≅ ∠2 m∠1 = m∠2 m∠2 = m∠1 ∠2 ≅ ∠1 Virginia Department of Education ©2013 Reasons Given Definition of congruent angles Symmetric Property of Equality Definition of congruent angles Geometry Vocabulary Cards Page 16 Properties of Congruence Reflexive Property For all angles A, ∠A ≅ ∠A. An angle is congruent to itself. For any angles A and B, Symmetric Property If ∠A ≅ ∠B, then ∠B ≅ ∠A . Order of congruence does not matter. For any angles A, B, and C, If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C. Transitive Property If two angles are both congruent to a third angle, then the first two angles are also congruent. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 17 Law of Detachment deductive reasoning stating that if the hypothesis of a true conditional statement is true then the conclusion is also true 120° A Example: If m∠A > 90°, then ∠A is an obtuse angle. m∠A = 120°. Therefore, ∠A is an obtuse angle. If p→q is a true conditional statement and p is true, then q is true. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 18 Law of Syllogism deductive reasoning that draws a new conclusion from two conditional statements when the conclusion of one is the hypothesis of the other Example: 1. If a rectangle has four equal side lengths, then it is a square. 2. If a polygon is a square, then it is a regular polygon. 3. If a rectangle has four equal side lengths, then it is a regular polygon. If p→q and q→r are true conditional statements, then p→r is true. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 19 Counterexample specific case for which a conjecture is false Example: Conjecture: “The product of any two numbers is odd.” Counterexample: 2 ∙ 3 = 6 One counterexample proves a conjecture false. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 20 Perpendicular Lines two lines that intersect to form a right angle m n Line m is perpendicular to line n. m⊥n Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 21 Parallel Lines lines that do not intersect and are coplanar m n m||n Line m is parallel to line n. Parallel lines have the same slope. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 22 Skew Lines lines that do not intersect and are not coplanar m n Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 23 Transversal a line that intersects at least two other lines t t a x y b Line t is a transversal. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 24 Corresponding Angles angles in matching positions when a transversal crosses at least two lines t 1 2 3 4 a 6 5 7 8 b Examples: 1) ∠2 and ∠6 2) ∠3 and ∠7 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 25 Alternate Interior Angles angles inside the lines and on opposite sides of the transversal t 1 2 a 3 4 b Examples: 1) ∠1 and ∠4 2) ∠2 and ∠3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 26 Alternate Exterior Angles angles outside the two lines and on opposite sides of the transversal 1 t 2 a 3 4 b Examples: 1) ∠1 and ∠4 2) ∠2 and ∠3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 27 Consecutive Interior Angles angles between the two lines and on the same side of the transversal t 1 3 a 2 4 b Examples: 1) ∠1 and ∠2 2) ∠3 and ∠4 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 28 Parallel Lines t 1 3 2 a 4 5 7 6 8 b Line a is parallel to line b when Corresponding angles are congruent Alternate interior angles are congruent Alternate exterior angles are congruent Consecutive interior angles are supplementary Virginia Department of Education ©2013 ∠1 ≅ ∠5, ∠2 ≅ ∠6, ∠3 ≅ ∠7, ∠4 ≅ ∠8 ∠3 ≅ ∠6 ∠4 ≅ ∠5 ∠1 ≅ ∠8 ∠2 ≅ ∠7 m∠3+ m∠5 = 180° m∠4 + m∠6 = 180° Geometry Vocabulary Cards Page 29 Midpoint divides a segment into two congruent segments C D M Example: M is the midpoint of CD CM ≅ MD CM = MD Segment bisector may be a point, ray, line, line segment, or plane that intersects the segment at its midpoint. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 30 Midpoint Formula given points A(x1, y1) and B(x2, y2) midpoint M = B (x2, y2) M A Virginia Department of Education ©2013 (x1, y1) Geometry Vocabulary Cards Page 31 Slope Formula ratio of vertical change to horizontal change change in y y2 – y1 slope = m = = change in x x2 – x1 x2 – x1 y2 – y1 A Virginia Department of Education ©2013 B (x2, y2) (x1, y1) Geometry Vocabulary Cards Page 32 Slopes of Lines Parallel lines have the same slope. Perpendicular lines have slopes whose product is -1. y n p x Vertical lines have undefined slope. Horizontal lines have 0 slope. Example: The slope of line n = -2. The slope of line p = 1 . -2 ∙ 1 = -1, therefore, n ⊥ p. 2 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 33 2 Distance Formula given points A (x1, y1) and B (x2, y2) AB = ( x2 − x1 ) 2 + ( y 2 − y1 ) x2 – x1 y2 – y1 A 2 B (x2, y2) (x1, y1) The distance formula is based on the Pythagorean Theorem. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 34 Line Symmetry MOM B X Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 35 Point Symmetry Cˊ A P Aˊ C pod S Z Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 36 Rotation y B A C A′ B′ D′ C′ D x center of rotation Preimage A(-3,0) B(-3,3) C(-1,3) D(-1,0) Image A′(0,3) B′(3,3) C′(3,1) D′(0,1) Pre-image has been transformed by a 90° clockwise rotation about the origin. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 37 Rotation y A A' x center of rotation Pre-image A has been transformed by a 90° clockwise rotation about the point I (2, 0) to form image A . Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 38 Reflection y F F′ x E′ D′ Preimage D(1,-2) E(3,-2) F(3,2) Virginia Department of Education ©2013 D E Image D′(-1,-2) E′(-3,-2) F′(-3,2) Geometry Vocabulary Cards Page 39 Translation y D E C A B x E′ D′ C′ A′ B′ Preimage A(1,2) B(3,2) C(4,3) D(3,4) E(1,4) Virginia Department of Education ©2013 Image A′(-2,-3) B′(0,-3) C′(1,-2) D′(0,-1) E′(-2,-1) Geometry Vocabulary Cards Page 40 Dilation y A′ A C C′ B B′ x Preimage A(0,2) B(2,0) C(0,0) Image A′(0,4) B′(4,0) C′(0,0) F′ E′ Preimage E F G H Image E′ F′ G′ H′ F E H′ H G P Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 41 G′ Perpendicular Bisector a segment, ray, line, or plane that is perpendicular to a segment at its midpoint Z X s M Y Example: Line s is perpendicular to XY. M is the midpoint, therefore XM ≅ MY. Z lies on line s and is equidistant from X and Y. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 42 Constructions Traditional constructions involving a compass and straightedge reinforce students’ understanding of geometric concepts. Constructions help students visualize Geometry. There are multiple methods to most geometric constructions. These cards illustrate only one method. Students would benefit from experiences with more than one method and should be able to justify each step of geometric constructions. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 43 Construct segment CD congruent to segment AB Fig. 1 Fig. 2 A B C D Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 44 Construct a perpendicular bisector of segment AB B A Fig. 1 B A Fig. 2 B A Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 45 Construct a perpendicular to a line from point P not on the line P P B A B A Fig. 2 Fig. 1 P A P B B A Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 46 Construct a perpendicular to a line from point P on the line A P B A Fig. 1 A B P Fig. 2 P B A P Fig. 4 Fig. 3 Virginia Department of Education ©2013 B Geometry Vocabulary Cards Page 47 Construct a bisector of ∠A A A Fig. 1 A Fig. 2 A Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 48 Construct ∠Y congruent to ∠A A A Y Fig. 1 Y A A Y Y Fig. 2 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 49 Construct line n parallel to line m through point P not on the line P m Fig. 1 P Draw a line through point P intersecting line m. P m m Fig. 2 P n m Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 50 Construct an equilateral triangle inscribed in a circle Fig. 2 Fig. 1 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 51 Construct a square inscribed in a circle Fig. 1 Draw a diameter. Fig. 2 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 52 Construct a regular hexagon inscribed in a circle Fig. 2 Fig. 1 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 53 Construct the inscribed circle of a triangle Fig. 1 Bisect all angles. Fig. 2 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 54 Construct the circumscribed circle of a triangle Fig. 2 Fig. 1 Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 55 Construct a tangent from a point outside a given circle to the circle P P Fig. 2 Fig. 1 P P Fig. 4 Fig. 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 56 Triangles Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 57 Classifying Triangles Scalene Isosceles Equilateral No congruent sides At least 2 congruent sides 3 congruent sides No congruent angles 2 or 3 congruent angles 3 congruent angles All equilateral triangles are isosceles. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 58 Classifying Triangles Acute Right Obtuse Equiangular 3 acute angles 1 right angle 1 obtuse angle 3 angles, 1 angle 1 angle each less equals 90° greater than 90° than 90° Virginia Department of Education ©2013 Geometry Vocabulary Cards 3 congruent angles 3 angles, each measures 60° Page 59 Triangle Sum Theorem B A C measures of the interior angles of a triangle = 180° m∠A + m∠B + m∠C = 180° Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 60 Exterior Angle Theorem B 1 A C Exterior angle, m∠1, is equal to the sum of the measures of the two nonadjacent interior angles. m∠1 = m∠B + m∠C Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 61 Pythagorean Theorem C a B b c hypotenuse A If ∆ABC is a right triangle, then 2 2 2 a +b =c . Conversely, if a2 + b2 = c2, then ∆ABC is a right triangle. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 62 Angle and Side Relationships ∠A is the largest angle, therefore BC is the longest side. B A 88o 8 38o 6 54o 12 A 88o 8 B 38 6 54o o 12 Virginia Department of Education ©2013 C ∠B is the smallest angle, therefore AC is the shortest side. Geometry Vocabulary Cards Page 63 C Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. B A C Example: AB + BC > AC AC + BC > AB AB + AC > BC Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 64 Congruent Triangles E B F A D C Two possible congruence statements: ∆ABC ≅ ∆FED ∆BCA ≅ ∆EDF Corresponding Parts of Congruent Figures ∠A ≅ ∠F AB ≅ FE ∠B ≅ ∠E BC ≅ ED ∠C ≅ ∠D CA ≅ DF Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 65 B SSS Triangle Congruence Postulate A E F D C Example: If Side AB ≅ FE, Side AC ≅ FD, and Side BC ≅ ED , then ∆ ABC ≅ ∆FED. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 66 SAS Triangle Congruence Postulate B E C A F D Example: If Side AB ≅ DE, Angle ∠A ≅ ∠D, and Side AC ≅ DF , then ∆ ABC ≅ ∆DEF. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 67 HL Right Triangle Congruence S R T Y X Z Example: If Hypotenuse RS ≅ XY, and Leg ST ≅ YZ , then ∆ RST ≅ ∆XYZ. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 68 ASA Triangle Congruence Postulate B E F C A D Example: If Angle ∠A ≅ ∠D, Side AC ≅ DF , and Angle ∠C ≅ ∠F then ∆ ABC ≅ ∆DEF. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 69 AAS Triangle Congruence Theorem S R T Y Z X Example: If Angle ∠R ≅ ∠X, Angle ∠S ≅ ∠Y, and Side ST ≅ YZ then ∆ RST ≅ ∆XYZ. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 70 Similar Polygons C B 4 A F G D 12 H 6 E 2 ABCD ∼ HGFE Angles ∠A corresponds to ∠H ∠B corresponds to ∠G ∠C corresponds to ∠F ∠D corresponds to ∠E Sides corresponds to corresponds to corresponds to corresponds to Corresponding angles are congruent. Corresponding sides are proportional. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 71 Similar Polygons and Proportions B 6 C G 4 F x 12 H A Corresponding vertices are listed in the same order. Example: ∆ABC ∼ ∆HGF AB HG BC = GF 12 6 = x 4 The perimeters of the polygons are also proportional. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 72 AA Triangle Similarity Postulate S Z X R Example: Y T If Angle ∠R ≅ ∠X and Angle ∠S ≅ ∠Y, then ∆RST ∼ ∆XYZ. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 73 SAS Triangle Similarity Theorem B E 14 7 C A Example: 12 D F 6 If ∠A ≅ ∠D and AB AC = DE DF then ∆ABC ∼ ∆DEF. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 74 SSS Triangle Similarity Theorem S 13 R Example: Y 6.5 5 T 12 X 2.5 6 Z RT RS ST If = = XZ XY YZ then ∆RST ∼ ∆XYZ. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 75 Altitude of a Triangle a segment from a vertex perpendicular to the opposite side G G A altitudes orthocenter B J B P IH C altitude/height Every triangle has 3 altitudes. The 3 altitudes intersect at a point called the orthocenter. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 76 Median of a Triangle C median A D B D is the midpoint of AB; therefore, CD is a median of ∆ABC. Every triangle has 3 medians. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 77 Concurrency of Medians of a Triangle A centroid D C P E B F Medians of ∆ABC intersect at P and 2 2 2 AP = AF, CP = CE , BP = BD. 3 Virginia Department of Education ©2013 3 Geometry Vocabulary Cards 3 Page 78 30°-60°-90° Triangle Theorem x 60° 2x x 3 x 30° 30° 60° Given: short leg = x Using equilateral triangle, hypotenuse = 2 ∙ x Applying the Pythagorean Theorem, longer leg = x ∙ 3 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 79 45°-45°-90° Triangle Theorem 45° x 2 x x 45° Given: leg = x, then applying the Pythagorean Theorem; 2 2 2 hypotenuse = x + x hypotenuse = x 2 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 80 Geometric Mean of two positive numbers a and b is the positive number x that satisfies a x = . x b 2 x = ab and x = ab. In a right triangle, the length of the altitude is the geometric mean of the lengths of the two segments. A x B 9 4 C Example: 2 = , so x = 36 and x = 36 = 6. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 81 Trigonometric Ratios B (hypotenuse) c a (side opposite ∠A) A b (side adjacent ∠A) C sin A = side opposite ∠A = a c hypotenuse cos A = side adjacent ∠A = b c hypotenuse tan A = side opposite ∠A = a side adjacent to ∠A b Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 82 Inverse Trigonometric Ratios B A c a b C Definition Example If tan A = x, then tan-1 x = m∠A. tan-1 a = m∠A b If sin A = y, then sin-1 y = m∠A. sin-1 a = m∠A c If cos A = z, then cos-1 z = m∠A. cos-1 b = m∠A c Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 83 Area of Triangle B a C h A b sin C = h a h = a∙sin C A= 1 bh (area of a triangle formula) 2 1 By substitution, A = b(a∙sin C) 2 1 A = ab∙sin C 2 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 84 Polygons and Circles Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 85 Polygon Exterior Angle Sum Theorem The sum of the measures of the exterior angles of a convex polygon is 360°. 1 2 5 4 3 Example: m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360° Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 86 Polygon Interior Angle Sum Theorem The sum of the measures of the interior angles of a convex n-gon is (n – 2)∙180°. S = m∠1 + m∠2 + … + m∠n = (n – 2)∙180° 1 2 5 3 4 Example: If n = 5, then S = (5 – 2)∙180° S = 3 ∙ 180° = 540° Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 87 Regular Polygon a convex polygon that is both equiangular and equilateral Equilateral Triangle Each angle measures 60o. Square Each angle measures 90o. Regular Pentagon Each angle measures 108o. Regular Hexagon Each angle measures 120o. Regular Octagon Each angle measures 135o. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 88 Properties of Parallelograms • Opposite sides are parallel and congruent. • Opposite angles are congruent. • Consecutive angles are supplementary. • The diagonals bisect each other. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 89 Rectangle • A rectangle is a parallelogram with four right angles. • Diagonals are congruent. • Diagonals bisect each other. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 90 Rhombus • A rhombus is a parallelogram with four congruent sides. • Diagonals are perpendicular. • Each diagonal bisects a pair of opposite angles. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 91 Square • A square is a parallelogram and a rectangle with four congruent sides. • Diagonals are perpendicular. • Every square is a rhombus. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 92 Trapezoid • A trapezoid is a quadrilateral with exactly one pair of parallel sides. • Isosceles trapezoid – A trapezoid where the two base angles are equal and therefore the sides opposite the base angles are also equal. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 93 Circle all points in a plane equidistant from a given point called the center • Point O is the center. • MN passes through the center O and therefore, MN is a diameter. • OP, OM, and ON are radii and OP ≅ OM ≅ ON. • RS and MN are chords. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 94 Circles A polygon is an inscribed polygon if all of its vertices lie on a circle. A circle is considered “inscribed” if it is tangent to each side of the polygon. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 95 Circle Equation y (x,y) r y x x 2 2 2 x +y =r circle with radius r and center at the origin standard equation of a circle (x – h)2 + (y – k)2 = r2 with center (h,k) and radius r Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 96 Lines and Circles C A D B • Secant (AB) – a line that intersects a circle in two points. • Tangent (CD) – a line (or ray or segment) that intersects a circle in exactly one point, the point of tangency, D. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 97 Secant y° 1 x° If two lines intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measures of the intercepted arcs. 1 m∠1 = (x° + y°) 2 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 98 Tangent Q S R A line is tangent to a circle if and only if the line is perpendicular to a radius drawn to the point of tangency. QS is tangent to circle R at point Q. Radius RQ ⊥ QS Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 99 Tangent C A B If two segments from the same exterior point are tangent to a circle, then they are congruent. AB and AC are tangent to the circle at points B and C. Therefore, AB ≅ AC and AC = AB. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 100 Central Angle an angle whose vertex is the center of the circle B minor arc AB D C major arc ADB A ∠ACB is a central angle of circle C. Minor arc – corresponding central angle is less than 180° Major arc – corresponding central angle is greater than 180° Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 101 Measuring Arcs B C 110° 70° R A Minor arcs Major arcs D Semicircles The measure of the entire circle is 360o. The measure of a minor arc is equal to its central angle. The measure of a major arc is the difference between 360° and the measure of the related minor arc. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 102 Arc Length B 4 cm 120° C A Example: Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 103 Secants and Tangents Secant-tangent Two secants x° 1 y° x° 1 y° Two tangents x° 1 y° m∠1 = Virginia Department of Education ©2013 1 (x°- y°) 2 Geometry Vocabulary Cards Page 104 Inscribed Angle angle whose vertex is a point on the circle and whose sides contain chords of the circle B A C Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 105 Area of a Sector region bounded by two radii and their intercepted arc Example: cm Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 106 Inscribed Angle Theorem D B A C If two inscribed angles of a circle intercept the same arc, then the angles are congruent. ∠BDC ≅ ∠BAC Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 107 Inscribed Angle Theorem B A O C m∠BAC = 90° if and only if BC is a diameter of the circle. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 108 Inscribed Angle Theorem T 92° 95° H J A 85° 88° M M, A, T, and H lie on circle J if and only if m∠A + m∠H = 180° and m∠T + m∠M = 180°. Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 109 Segments in a Circle c a b d If two chords intersect in a circle, then a∙b = c∙d. Example: 12(6) = 9x 72 = 9x 8=x Virginia Department of Education ©2013 x 12 Geometry Vocabulary Cards 6 9 Page 110 Segments of Secants Theorem A B D C E AB ∙ AC = AD ∙ AE Example: 6 9 x 16 Virginia Department of Education ©2013 6(6 + x) = 9(9 + 16) 36 + 6x = 225 x = 31.5 Geometry Vocabulary Cards Page 111 Segments of Secants and Tangents Theorem A B C E 2 AE = AB ∙ AC Example: 252 = 20(20 + x) 625 = 400 + 20x x = 11.25 Virginia Department of Education ©2013 20 25 Geometry Vocabulary Cards x Page 112 Three-Dimensional Figures Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 113 Cone solid that has a circular base, an apex, and a lateral surface apex lateral surface (curved surface of cone) slant height (l) height (h) radius(r) base 1 2 V = πr h 3 L.A. (lateral surface area) = πrl 2 S.A. (surface area) = πr + πrl Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 114 Cylinder solid figure with congruent circular bases that lie in parallel planes base height (h) base radius (r) 2 V = πr h L.A. (lateral surface area) = 2πrh 2 S.A. (surface area) = 2πr + 2πrh Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 115 Polyhedron solid that is bounded by polygons, called faces Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 116 Similar Solids Theorem If two similar solids have a scale factor of a:b, then their corresponding surface areas have a ratio of a2: b2, and their corresponding volumes have a ratio of a3: b3. cylinder A ∼ cylinder B Example A B Virginia Department of Education ©2013 scale factor a:b 3:2 ratio of surface areas a2: b2 9:4 ratio of volumes a3: b3 27:8 Geometry Vocabulary Cards Page 117 Sphere a three-dimensional surface of which all points are equidistant from a fixed point radius 4 3 V = πr 3 S.A. (surface area) = 4πr Virginia Department of Education ©2013 Geometry Vocabulary Cards 2 Page 118 Pyramid polyhedron with a polygonal base and triangular faces meeting in a common vertex vertex slant height (l) height (h) area of base (B) base perimeter of base (p) 1 V = Bh 3 1 L.A. (lateral surface area) = lp 2 1 S.A. (surface area) = lp + B 2 Virginia Department of Education ©2013 Geometry Vocabulary Cards Page 119